Is This Linear Algebra Problem Set Correct?

[will]

i did most of them,, u guys are prob experts I am having big probelm with the a few of them
please check my work, thanks
hints please.,

True or False, only True if it is always true
1) Any linearly independent set of three vectors in R^3 is a basis for R^3___true

2) Any set of five vectors in R^4 spans R^4__false

3) Any set of four vectors in R^3 is linearly independent__false

4) the set {(1,0,1),(-5,4,-9),(5,-3,8),(2,-1,3)} spans R^3___

5) the set {x^3 – x + 2, x^2 + x – 2, 3x^3 + 2x^2 + 4x} spans P^3___

6) If a vector space V has a basis S with 7 elements, then any other basis T for V also has

7 elements____true

7) If a set S of vectors in V contains the zero vector, then S is linearly dependent___true

8) If dim(V) = n, then any set of n – 1 vectors in V must be linearly independent___

9) if dim(V) = n, then any set of n + 1 vectors in V must be linearly independent____

10) if dim(V) = n, then there exists a set of n + 1 vectors in V that spans V____

 

[zhentil]

On #4, they'll span R^3 if 3 of them are linearly independent, correct? On most of these true/false about "all vectors in X must be..." choosing all zero vectors usually provides a counterexample. And on 10, if dim(V) = n, the basis for V + 0-vector would provide the example, correct?

 

[radou]

5) the set {x^3 – x + 2, x^2 + x – 2, 3x^3 + 2x^2 + 4x} spans P^3___

It is not hard to show that, for a vector space of dimension n, no set with less then n elements can span that vector space.

8) If dim(V) = n, then any set of n – 1 vectors in V must be linearly independent___

Let V = R^3. Let S = {(1, 0, 0), (2, 0, 0)}. Is S independent?

10) if dim(V) = n, then there exists a set of n + 1 vectors in V that spans V____

What do you think? What is the exact definition of "span"? Is there any restriction on the number of vectors in this definition?